A stochastic maximum principle for a stochastic differential game of a mean-field type

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A stochastic maximum principle for a stochastic differential game of a mean-field type

John Hosking

To cite this version:

John Hosking. A stochastic maximum principle for a stochastic differential game of a mean-field type.

2011. �hal-00641090v2�

A stochastic maximum principle for a stochastic differential game of a mean-field type ^∗

John J. A. Hosking

November 15, 2011

Abstract

We construct a stochastic maximum principle (SMP) which provides necessary conditions for the existence of Nash equilibria in a certain form ofN-agent stochastic differential game (SDG) of a mean- field type. The information structure considered for the SDG is of a possible asymmetric and partial type. To prove our SMP we use a spike-variation approach with adjoint representation techniques, analogous to that of S. Peng in the optimal stochastic control context. In our proof we apply adjoint representation procedures at three points. The first-order adjoint processes are defined as solutions to certain mean-field backward stochastic differential equations, and second-order adjoint processes of a first type are defined as solutions to certain backward stochastic differential equations. Second- order adjoint processes of a second type are defined as solutions of backward stochastic equations of a type that we introduce in this paper, and which we term conditional mean-field backward stochastic differential equations. From the resulting representations, we show that the terms relating to these second-order adjoint processes of the second type are of an order such that they do not appear in our final SMP equations.

Keywords: Stochastic maximum principle, stochastic differential games, Nash equilibria, mean- field type stochastic equations, mean-field type backward stochastic equations.

American Mathematical Society, 2010 Mathematics Subject Classification: 93E99, 60H99, 91A06, 91A15.

1 Introduction

In this paper we construct a stochastic maximum principle (SMP) which provides necessary conditions for the existence of Nash equilibria in a certainN-agent stochastic differential game (SDG) of a mean- field type. The exact description of this form ofN-agent SDG is presented in Section 2; in summary, it is a SDG with:

• a state process of the form X^(u)(ω, t) =x0+

Zt 0

b

ω, s, X^(u)(ω, s), Z

Ω

φb(ω⁰, s, X^(u)(ω⁰, s), u(ω⁰, s))P(dω⁰), u(ω, s)

ds +

Z t 0

σ

ω, s, X^(u)(ω, s), Z

Ω

φσ(ω⁰, s, X^(u)(ω⁰, s), u(ω⁰, s))P(dω⁰), u(ω, s)

dB(ω, s) for eachu= (u1, . . . , uN) comprising of each agent’s employed admissible control process; and

• performance functionals{Ji|i∈ {1, . . . , N}}of the form Ji(u) :=

Z

Ω

hZ T 0

fi

ω, s, X^(u)(ω, s), Z

Ω

φf_i(ω⁰, s, X^(u)(ω⁰, s), u(ω⁰, s))P(dω⁰), u(ω, s) ds +gi

ω, X^(u)(ω, T), Z

Ω

φg_i(ω⁰, X^(u)(ω⁰, T))P(dω⁰)i P(dω).

∗A version of this paper which addresses a multidimensional form of the problem that is considered here, has been deposited on HAL - INRIA (see [10])

†INRIA Paris - Rocquencourt, ´Equipe - projet MATHFI, Domaine de Voluceau, BP 105, 78153 Le Chesnay Cedex, France (John.Hosking@inria.fr)

Key features of the problem we address are that:

• The setting is in general non-Markovian.

• The information structure is in general of an asymmetric and partial type, that is:

– For each agent, their class of admissible controls is a class of predictable processes with respect to a given agent-specific filtrationGⁱ(which are not necessarily equal to one another).

– All of the agent’s filtrations,{Gⁱ|i∈ {1, . . . , N}}, are subfiltrations of a reference filtration F.

• For each agent, their admissible control processes are valued in a given agent-specific complete separable metric space—which is not necessarily convex.

We construct our SMP using a spike variation approach with adjoint representation techniques, analogous to that of Peng [21] where a SMP is constructed for a certain form of stochastic optimal control problem in a non-mean-field setting. The approach we follow here is also analogous, in certain respects, to that in the works of [19] and [5], in which the original work of Peng [21] is further devel- oped in the directions of stochastic control problems with multidimensional performance functionals (including the issue of Nash equilibria in SDGs), and a mean-field type stochastic optimal control problem, respectively.

In [19] the authors develop an approach that is to some extent different to the approach of Peng [21], and they allow for the case of multidimensional performance functionals as well as for classes of admissible control processes that satisfy a weaker integrability condition then that stated in [21]. A SMP providing necessary conditions for Nash equilibria in a certain N-agent SDG—with a standard information structure and not of a mean-field type—is presented in Theorem 5.2 of [19].

In [5] a Peng-type SMP is constructed for a certain stochastic optimal control problem of a mean- field type. The generality of the SDG problem that we consider in this paper and that of the stochastic optimal control problem considered in [5] is different. In particular, we allow for the possibility of an asymmetric and partial information structure for our SDG. Moreover, a fundamental difference between these two works is the manner in which the limits relating to the respective second form of quadratic-type terms, as we refer to them (see the proof of Theorem 3.6 for further explanation), are calculated. In [5] the relevant limit is implied to be zero by establishing an appropriate upper bound for the modulus of those terms. In this work we calculate the relevant limit by applying the adjoint representation procedure a third time, defining the relevant adjoint processes as solutions to a backward stochastic equation of a type that we introduce in this paper, and which we term condition mean-field backward stochastic differential equations (CMFBSDEs). We propose that our approach to deal with the second form of quadratic-type terms is no more complicated than that of [5].

Since, having introduced the concept of CMFBSDEs, then our approach to dealing with these terms is essentially the application of some of the same general ideas, such as the adjoint representation techniques, from [21], which we also make use of in treating, what we refer to as, the linear-type termsand thefirst form of quadratic-type terms in our problem here. Also, it would appear that our approach may—in principle—be suitable for further development in an attempt to calculate higher- order expansion terms for the second form of quadratic-terms, which would be of interest in the possible construction of forms of higher-order SMPs.

As is suggested by the form of the state processX^(u), and of the performance functionals{Ji|i∈ {1, . . . , N}}, stated above, the termmean-field is broadly used here in the sense which takes:

• a stochastic process of a mean-field type to be a form of stochastic process whose dynamics are a function of its time-marginal probability laws;

• a stochastic control problem of a mean-field type, or a SDG of a mean-field type, to be a stochastic control problem or a SDG where the state process is of a mean-field type (in the above sense) and/or where the performance functionals are of a mean-field type—in the sense that: for a performance functional given as an expectation of the sum of a running performance term and a terminal performance term, then either of these two terms are themselves functions of the time- marginal probability laws of the state process (and not both in a manner that can be reduced to the standard case).

This use of the term mean-field is collectively motivated by its uses in, for example, [7], [6], [5], [1]

and [18].

A notion of a mean-field backward stochastic differential equation (MFBSDE) appears in [6] and [7]:

this notion of a MFBSDE is essentially a generalization of that of a BSDE which allows the generator term to be a function of the time-marginal probability laws of the solution. Related to the subject of SDEs of a mean-field type, in this sense, is a large literature on approximations by interacting

particle systems, see for example [17], [4], and [11]. For an example of a stochastic control problem with McKean-Vlasov type SDEs, see [3].

In [1] a SMP is constructed for a mean-field stochastic control problem where both the state dy- namics and the performance functionals are of a mean-field type. A fundamental difference between [1]

and both this work as well as [5], is that in [1] the construction of their SMP is carried out by an extension of the Bensoussan [2] approach to a mean-field setting; whereas in this paper, and in [5], the SMPs are constructed by extending the Peng [21] approach to a mean-field setting (although with the differences mentioned above). That is, the setting in [1] allows for the admissible controlsuto be perturbed in the manneru+¯u, given some admissible control ¯u. Our problem of interest is not to permit this form of perturbation; we do not assume the required structure on the range of the control processes for this form of perturbation to be valid, and instead we work with perturbations given by spike-variations. Also, note that in this paper we allow the controls to appear directly—that is other than just via their effect on the state process itself—in the mean-field components of the state process and performance functionals: which is not the case for the control processes in [1] or in [5]. That is, in a certain form, the dynamics and performance functionals in our SDG may be functions of the time-marginal joint probability laws between the state process and the control processes.

Another relevant work is that of [18] which constructs, via results from Malliavin calculus, a SMP for a stochastic control problem in which mean-field terms appear in the considered form of performance functional but not in the dynamics of the state process (which is a controlled Itˆo-L´evy SDE). By considering a certain form of Bensoussan-type perturbation, the authors make use of the relevant Malliavin calculus duality relations to construct a SMP in terms of the Malliavin derivative and difference operators.

Other work concerned with a notion of mean-field in the area of optimal control and game theory is that of the series of work by J.-M. Lasry and P.-L. Lions, for example [14], [15] and [16]. See also [9].

Mean-field games are the subject of, for example, Section 2 of [16] where a certain form ofN-agent SDG (with non-mean-field state process and performance functionals, in the sense from before) and certain corresponding equations for the existence of Nash equilibria are considered as the number of agentsN tends to infinity. At first sight this form of problem appears to be conceptually different to the problem we consider here, the later being a finite-agent SDG with a mean-field type state process and performance functionals. However, it could be a topic for future research to consider whether there exists some form of relationship between mean-field SDGs of the type addressed here, or some variant of them, and a general notion of mean-field games.

This paper is structured as follows: Section 2 specifies the form of the SDG that we consider, presenting the definitions of the admissible control classes, the state dynamics, and the performance functionals; Section 3 presents the main result of the paper, Theorem 3.6, which presents our SMP for Nash equilibria in our considered SDG; Section 4 provides an existence and uniqueness result for solutions to certain CMFBSDEs; Section 5 presents a brief conclusion; and Appendix A contains a certain auxiliary result.

2 Setting

This section details the form of the SDG for which our SMP will be constructed. For the sake of simplicity we consider the problem in its reduced one-dimensional form: that is we consider the driving Brownian motion, the state process and its coefficient functions{b, σ, φb, φσ}, and all the functions in each agent’s performance functional{fi, gi, φf_i, φgi|i∈ {1, . . . , N}}as being valued inR. A version of this paper which addresses the more general multidimensional form of the problem is [10].

Fix anyT ∈(0,∞) andx0∈R. For eachi∈ {1, . . . , N}let (Ui, di) be a complete separable metric space, and denoteU:=QN

i=1Ui. Let (Ω,F,P) be a complete probability space, andB: Ω×[0, T]→R a standard Brownian motion with respect to its augmented natural filtrationF:={Ft|t∈[0, T]}(that is known to be right-continuous), and whichP-almost surely is taken as having continuous paths. It is assumed thatFT =F. Given any filtration O={Ot|t∈[0, T]}for (Ω,F,P), then we denote by P(O) the corresponding predictableσ-algebra on Ω×[0, T] with respect toO.

We assume throughout that the collection of coefficient functions has the properties detailed in Assumption 2.1 below.

Assumption 2.1. The functions











{b, σ, fi: Ω×[0, T]×R²×U →R|i∈ {1, . . . , N}}

{φb, φσ, φf_i: Ω×[0, T]×R×U→R|i∈ {1, . . . , N}}

{gi: Ω×R²→R|i∈ {1, . . . , N}}

{φg_i: Ω×R→R|i∈ {1, . . . , N}}

have the following properties:

(i)

b, σ,{fi|i∈ {1, . . . , N}} are P(F)× B(R²×U)|B(R)-measurable;

φb, φσ,{φf_i|i∈ {1, . . . , N}} are P(F)× B(R×U)|B(R)-measurable;

{gi|i∈ {1, . . . , N}} are F × B(R²)|B(R)-measurable;

{φg_i|i∈ {1, . . . , N}} are F × B(R)|B(R)-measurable.

(ii) ForP-almost allω∈Ω, the functions:

(x, y)7→b(ω, t, x, y, u), σ(ω, t, x, y, u), fi(ω, t, x, y, u) are in C²(R²;R), x7→φb(ω, t, x, u), φσ(ω, t, x, u), φf_i(ω, t, x, u) are in C²(R;R),

(x, y)7→gi(ω, x, y) are in C²(R²;R), x7→φg_i(ω, x) are in C²(R;R), uniformly for anyt∈[0, T]andu∈U, for alli∈ {1, . . . , N}.

(iii) There exists a constantc >0such that:

max n

|ψ(ω, t,0,0, u)|,|∂xψ(ω, t, x, y, u)|,|∂yψ(ω, t, x, y, u)|,|∂x,x⁽²⁾ψ(ω, t, x, y, u)|,

|∂x,y⁽²⁾ψ(ω, t, x, y, u)|,|∂y,y⁽²⁾ψ(ω, t, x, y, u)|,|φψ(ω, t,0, u)|,

|∂φψ(ω, t, x, u)|,|∂⁽²⁾φψ(ω, t, x, u)|

ψ∈ {b, σ, fi|i∈ {1, . . . , N}o

≤c

max n

|∂x,x⁽²⁾ψ(ω, t, x, y, u)−∂x,x⁽²⁾ψ(ω, t, x⁰, y⁰, u)|,|∂x,y⁽²⁾ψ(ω, t, x, y, u)−∂x,y⁽²⁾ψ(ω, t, x⁰, y⁰, u)|,

|∂y,y⁽²⁾ψ(ω, t, x, y, u)−∂⁽²⁾_y,yψ(ω, t, x⁰, y⁰, u)|,|∂⁽²⁾φψ(ω, t, x, u)−∂⁽²⁾φψ(ω, t, x⁰, u)|

ψ∈ {b, σ, fi|i∈ {1, . . . , N}o

≤c

|x−x⁰|+|y−y⁰|

max n

|∂xgi(ω, x, y)|,|∂ygi(ω, x, y)|,|∂⁽²⁾x,xgi(ω, x, y)|,|∂x,y⁽²⁾gi(ω, x, y)|,

|∂y,y⁽²⁾gi(ω, x, y)|,|∂φg_i(ω, x)|,|∂⁽²⁾φg_i(ω, x)|

i∈ {1, . . . , N}o

≤c

maxn

|∂x,x⁽²⁾gi(ω, x, y)−∂_x,x⁽²⁾gi(ω, x⁰, y⁰)|,|∂x,y⁽²⁾gi(ω, x, y)−∂⁽²⁾_x,ygi(ω, x⁰, y⁰)|,

|∂y,y⁽²⁾gi(ω, x, y)−∂_y,y⁽²⁾gi(ω, x⁰, y⁰)|,|∂⁽²⁾φg_i(ω, x)−∂⁽²⁾φg_i(ω, x⁰)|

i∈ {1, . . . , N}o

≤c

|x−x⁰|+|y−y⁰| forP-almost allω∈Ωuniformly for allt∈[0, T],x, x⁰, y, y⁰∈Randu∈U.

(iv) There exists a subsetΩ0 ⊆Ω withP(Ω0) = 1 such that for each ψ ∈ {b, σ, fi|i∈ {1, . . . , N}}

each mapu7→ψ(ω, t, x, y, u) andu7→φψ(ω, t, x, u)is continuous and bounded uniformly for all (ω, t)∈Ω0×[0, T]andx, y∈R.

Definition 2.2. Let{Gi|i∈ {1, . . . , N}}be a collection of—not necessarily equal—filtrations, where for eachi∈ {1, . . . , N}theGⁱ:={Gi,t|t∈[0, T]}is complete, right-continuous, and a subfiltration of F. For eachi∈ {1, . . . , N}, we defineAi, the class of admissible controls for thei^th-agent, as the set of allP(Gⁱ)|B(Ui)-measurable processesui: Ω×[0, T]→Ui.

Proposition 2.3 (cf. Theorem 4.1 in [7]). Under Assumption 2.1, for each u := (u1, . . . , uN) ∈ QN

i=1Ai there exists a unique—up to indistinguishability—stochastic processX^(u) : Ω×[0, T] →R thatP-almost surely has continuous paths and which:

• is a strongF-adapted solution to the following mean-field type SDE

X^(u)(t) =x0+ Z t

0

b

s, X^(u)(s), Z

Ω

φb(ω⁰, s, X^(u)(ω⁰, s), u(ω⁰, s))P(dω⁰), u(s) ds +

Zt 0

σ

s, X^(u)(s), Z

Ω

φσ(ω⁰, s, X^(u)(ω⁰, s), u(ω⁰, s))P(dω⁰), u(s)

dB(s) (1) forP-almost allω∈Ωuniformly for allt∈[0, T]; and

• is such that for eachp∈[1,∞)thenEP[sup_t∈[0,T]|X^(u)(t)|^p]<∞.

Proof. The result can be established by making suitable slight adaptations to the standard tech- niques for solutions to (Brownian motion driven) Lipschitz SDEs, in particular using a Picard-iteration method for the existence part of the result.

In this paper we focus solely on the form ofN-agent SDG that is specified by the state dynamics defined in the mean-field SDE (1) and by the performance functionals given in Definition 2.4 below.

Definition 2.4. Under Assumption 2.1, for each i∈ {1, . . . , N}define the performance functional Ji:QN

`=1A`→R, for thei^th-agent, by Ji(u) :=EP

hZ T 0

fi

t, X^(u)(t), Z

Ω

φf_i(ω⁰, t, X^(u)(ω⁰, t), u(ω⁰, t))P(dω⁰), u(t) dt +gi

X^(u)(T), Z

Ω

φg_i(ω⁰, X^(u)(ω⁰, T))P(dω⁰)i (2) for allu∈QN

`=1A`, where for eachu∈QN

`=1A` the controlled state processX^(u): Ω×[0, T]→Ris defined as the solution to the corresponding mean-field type SDE (1) in the sense of Proposition 2.3.

3 A Stochastic Maximum Principle

Recall that a Nash equilibrium, for the SDG detailed by equations (1) and (2), is an N-tuple of admissible controlsu^∗:= (u^∗1, . . . , u^∗_N)∈QN

`=1A`such that for eachi∈ {1, . . . , N}then Ji(u^∗)≥Ji(u^∗i(ui))

for allui∈ Ai, whereu^∗_i(ui) := (u^∗_i(ui)1, . . . , u^∗_i(ui)N)∈QN

`=1A`is defined by u^∗_i(ui)j:=

ui ifj=i;

u^∗_j for allj∈ {1, . . . , N}\{i}. (3) Assumption 3.1. Under Assumption 2.1, assume that there exists a u^∗= (u^∗1, . . . , u^∗_N)∈QN

`=1A`

which is a Nash equilibrium for the SDG of equations (1) and (2). LetX^∗: Ω×[0, T]→Rbe the state process corresponding to theN-tuple of admissible controlsu^∗,X^∗:=X^(u∗).

We make use of the following notation.

Notation 3.2. Suppose Assumptions 2.1 and 3.1 hold. For each functionψ∈ {b, σ, fi|i∈ {1, . . . , N}}

define∂ψ: Ω×[0, T]×R²×U×R²→Rand∂⁽²⁾ψ: Ω×[0, T]×R²×U×R³→Rby

∂ψ(ω, t, x, y, u, v, w) :=∂xψ(ω, t, x, y, u)v+∂yψ(ω, t, x, y, u)w

∂⁽²⁾ψ(ω, t, x, y, u, v, w, z) :=∂_x,x⁽²⁾ψ(ω, t, x, y, u)v²+ 2∂⁽²⁾_x,yψ(ω, t, x, y, u)vw +∂_y,y⁽²⁾ψ(ω, t, x, y, u)w²+∂yψ(ω, t, x, y, u)z

for P-almost all ω ∈ Ω uniformly for all t ∈ [0, T], x, y, v, w, z ∈ R, andu ∈ U. Define, for each i∈ {1, . . . , N}, the functions∂gi: Ω×R⁴→Rand∂⁽²⁾gi: Ω×R⁵→Rby

∂gi(ω, x, y, v, w) :=∂xgi(ω, x, y)v+∂ygi(ω, x, y)w

∂⁽²⁾gi(ω, x, y, v, w, z) :=∂_x,x⁽²⁾gi(ω, x, y)v²+ 2∂_x,y⁽²⁾gi(ω, x, y)vw+∂_y,y⁽²⁾gi(ω, x, y)w²+∂ygi(ω, x, y)z forP-almost allω∈Ωuniformly for allt∈[0, T], x, y, v, w, z∈R, andu∈U.

Define the shortened notation:

•

ψ(ω, t) :=ψ

ω, t, X^∗(ω, t), Z

Ω

φψ(ω⁰, t, X^∗(ω⁰, t), u^∗(ω⁰, t))P(dω⁰), u^∗(ω, t) for allψ∈ {λ, ∂xλ, ∂yλ, ∂x,x⁽²⁾λ, ∂x,y⁽²⁾λ, ∂y,y⁽²⁾λ|λ=b, σ, f1, . . . , fN};

•

ψ(ω, t) :=ψ

ω, t, X^∗(ω, t), u^∗(ω, t) for allψ∈ {λ, ∂λ, ∂⁽²⁾λ|λ=φb, φσ, φf1, . . . , φf_N};

•

ψ(ω) :=ψ

ω, X^∗(ω, T), Z

Ω

φψ(ω⁰, X^∗(ω⁰, T))P(dω⁰)

for allψ∈ {gi, ∂xgi, ∂ygi, ∂x,x⁽²⁾gi, ∂⁽²⁾x,ygi, ∂y,y⁽²⁾gi|i∈ {1, . . . , N}};

•

ψ(ω) :=ψ(ω, X^∗(ω, T)) for allψ∈ {φg_i, ∂φg_i, ∂⁽²⁾φg_i|i∈ {1, . . . , N}}.

Note that the definitions of the spacesS²(Ω×[0, T];R) andH²(Ω×[0, T];R) are given in Defini- tion 4.1.

Definition 3.3. Suppose Assumptions 2.1 and 3.1 hold. For each i ∈ {1, . . . , N} define the map hi: Ω×Ω×[0, T]×R⁴→Rby

hi(ω, ω⁰, t, v, v⁰, w, w⁰) :=∂xfi(ω, t) +∂yfi(ω⁰, t)∂φf_i(ω, t) +v∂xb(ω, t) +v⁰∂yb(ω⁰, t)∂φb(ω, t) +w∂xσ(ω, t) +w⁰∂yσ(ω⁰, t)∂φσ(ω, t) for all (ω, ω⁰) ∈ Ω0 ×Ω0, t ∈ [0, T], and v, v⁰, w, w⁰ ∈ R, for some Ω0 ⊆ Ω such that P(Ω0) = 1.

Then for eachi∈ {1, . . . , N}define the processes(pi, qi)∈ S²(Ω×[0, T];R)× H²(Ω×[0, T];R)as the solution of the following MFBSDE







pi(ω, t) =pi(ω, T) +RT t

R

Ωhi((ω, ω⁰), s, pi(ω, s), pi(ω⁰, s), qi(ω, s), qi(ω⁰, s))P(dω⁰) ds

−RT

t qi(ω, s)dB(ω, s) pi(ω, T) =∂xgi(ω) +EP[∂ygi]∂φgi(ω)

(4) forP-almost allω∈Ωuniformly for allt∈[0, T].

For eachi∈ {1, . . . , N}define the mapli: Ω×[0, T]×R²→Rby

li(t, v, w) :=∂_x,x⁽²⁾fi(t) +EP[∂yfi(t)]∂⁽²⁾φf_i(t) +pi(t)∂_x,x⁽²⁾b(t) +EP[pi(t)∂yb(t)]∂⁽²⁾φb(t)

+qi(t)∂⁽²⁾_x,xσ(t) +EP[qi(t)∂yσ(t)]∂⁽²⁾φσ(t) +v(2∂xb(t) + (∂xσ(t))²) + 2w∂xσ(t) forP-almost allω∈Ωuniformly for allt∈[0, T], andv, w∈R. Then for eachi∈ {1, . . . , N}define the processes(Pi, Qi)∈ S²(Ω×[0, T];R)× H²(Ω×[0, T];R) as the solution of the following BSDE

(

Pi(t) =Pi(T) +RT

t li(s, Pi(s), Qi(s)) ds−RT

t Qi(s) dB(s) Pi(T) =∂⁽²⁾x,xgi+EP[∂ygi]∂⁽²⁾φg_i

(5) forP-almost allω∈Ωuniformly for allt∈[0, T].

Remark 3.4. Note that given the assumed properties in Assumption 2.1 then:

• For eachi∈ {1, . . . , N}, the processes(pi, qi)in Definition 3.3 are well-defined, this can be shown by making a minor modification to Theorem 3.1 in [7].

• For each i∈ {1, . . . , N}, the processes (Pi, Qi) in Definition 3.3 are well-defined due to known results in the theory of BSDEs, see, for example, subsection 2.1 of [8].

Assumption 3.5. Suppose Assumptions 2.1 and 3.1 hold. Assume that, for each i ∈ {1, . . . , N}, there exist representations of the equivalence classes qi, Qi ∈ H²(Ω×[0, T];R) which are defined P- almost surely uniformly for allt∈[0, T]. In the following we take each qi andQi as denoting those representations.

Also, suppose that for eachi∈ {1, . . . , N}there exists a P(Gⁱ)× B(Ui)|B(R)-measurable process Hi: Ω×[0, T]×Ui→Rsuch that:

• the mapHi(ω, t,·) :Ui→Ris continuous and bounded, uniformly for all(ω, t)∈Ω×[0, T];

• for any givenui∈ Ai then Hi(ω, t, ui(ω, t)) =ρ_Gi

h fi

·, X^∗(·), Z

Ω

φf_i(ω⁰,·, X^∗(ω⁰,·), u^∗i(ui)(ω⁰,·))P(dω⁰), u^∗i(ui)(·) +pi(·)b

·, X^∗(·), Z

Ω

φb(ω⁰,·, X^∗(ω⁰,·), u^∗_i(ui)(ω⁰,·))P(dω⁰), u^∗_i(ui)(·) +qi(·)σ

·, X^∗(·), Z

Ω

φσ(ω⁰,·, X^∗(ω⁰,·), u^∗i(ui)(ω⁰,·))P(dω⁰), u^∗i(ui)(·) +Pi(·)

σ

·, X^∗(·), Z

Ω

φσ(ω⁰,·, X^∗(ω⁰,·), u^∗i(ui)(ω⁰,·))P(dω⁰), u^∗_i(ui)(·)

−σ

·, X^∗(·), Z

Ω

φσ(ω⁰,·, X^∗(ω⁰,·), u^∗(ω⁰,·))P(dω⁰), u^∗(·)2i (ω, t)

for P-almost all ω∈ Ω uniformly for all t∈ [0, T], where ρ_Gi[·]denotes the P(Gⁱ)-predictable projection.

Theorem 3.6 presents the SMP which provides necessary conditions for the existence of Nash equilibria in the considered SDG.

Theorem 3.6. Under Assumptions 2.1, 3.1 and 3.5, for u^∗∈QN

`=1A` to be a Nash equilibrium for the SDG of equations (1) and (2), as assumed, then it is necessary that for each i∈ {1, . . . , N} the following SMP is satisfied

maxv∈U_iHi(ω, t, v) =Hi(ω, t, u^∗i(ω, t)) (6) forP⊗Leb-almost all(ω, t)∈Ω×[0, T].

Proof. Fix any arbitraryi∈ {1, . . . , N}and any arbitrary admissible controlui∈ Aifor thei^th-agent.

Letu^∗i(u^(r,)_i ) = (u^∗i(u^(r,)_i )1, . . . , u^∗i(u^(r,)_i )N)∈QN

`=1A`denote u^∗i(u^(r,)_i )j:=

u^(r,)_i , ifj=i;

u^∗_j , ifj∈ {1, . . . , N}\{i};

where the family of spike-variations,{u^(r,)_i |r∈[0, T), ∈(0, T −r]}, of thei^th-agent’s controlu^∗i, is defined by

u^(r,)_i (t) :=u^∗i(t)(1−1(r,r+](t)) +ui(t)1(r,r+](t) for all (ω, t)∈Ω×[0, T]. We aim to show that

lim↓0⁻¹

Ji(u^∗_i(u^(r,)_i ))−Ji(u^∗)

=EP

h

Hi(r, ui(r))−Hi(r, u^∗_i(r))i

(7) for Leb-almost allr∈[0, T], since if this were true then under the assumption that theN-tupleu^∗of admissible controls is a Nash equilibrium, it would follow that for eachui∈ Ai then

EP

h

Hi(r, ui(r))−Hi(r, u^∗i(r))i

≤0 (8)

for Leb-almost allr∈[0, T]. This follows the general principle of the Peng [21] method when transfered to the SDG setting, and is analogous to the method used in [19] and that used in [5].

The main part of this proof is concerned with the calculation of the limit in equation (7). The ap- proach we take to do this is also developed on the general idea of that used in [21] for the corresponding problem. That approach being to expand the difference between the relevant two performance func- tionals into terms of certain types, and to introduce certain adjoint processes represented by backward stochastic equations (certain BSDEs in the case of [21]), to help one calculate the limit as required.

The approach we use is also related in certain respects to that in [19] and that in [5], which are themselves related, again in certain respects, to the original work of Peng [21].

Let{X^(r,): Ω×[0, T]→R|r∈[0, T), ∈(0, T−r]}denote the family of controlled state processes corresponding to the controlsu^∗_i(u^(r,)_i ). Also, let{ζ^(r,): Ω×[0, T]→R|r∈[0, T), ∈(0, T −r]}be the family of processes defined, for eachr∈[0, T) and∈(0, T −r], byζ^(r,):=X^(r,)−X^∗.

We use an expansion of the processζ^(r,)(see Lemma A.1), which, in regards to its format, is more akin to that in [20] than that which is used in [21] and [5], but which we use in a similar role. Given Lemma A.1 and Assumption 3.1, we then decompose the term Ji(u^∗i(u^(r,)_i ))−Ji(u^∗) into a sum of four components:

• a linear-type term with respect to the variablesζ^(r,);

• a quadratic-type term with respect to the variablesζ^(r,);

• a type of higher order term which we will show is of the ordero() as↓0; and

• a term for which the required limit may be calculated in a relatively simple manner;

An analogous form of decomposition may also be identified in [21] and [5]. The decomposition used in [19], that is in the sense of Lemma 3.1 of [19], is, to some extent, of different type of format.

Our linear-type term takes the form EP

hZ T r+

Y1(s)ζ^(r,)(s) ds+Y1(T)ζ^(r,)(T) i

for certain factors Y1. The procedure to deal with it is similar to that in [5] and is analogous to that which was first developed in [21], for their corresponding type terms. In [21], it was noted that

the relevant linear-type component is in fact a linear functional of another term, and that when this linear functional is written (by the Riesz representation theorem) as an inner-product between that other term and its relevant adjoint term, then one may calculate, as desired, the limit as↓0 of the linear-type term multiplied by⁻¹. With the expansion ofζ^(r,)that we use in this paper, we do not fully calculate the corresponding limit at this stage, but use the procedure to rewrite the linear-type term as a sum of elements that belong to the three other categories in the decomposition listed above.

In [21], it is also described how this relevant adjoint term may be given by the solution of a certain linear BSDE. In this paper we introduce all our adjoint processes directly as solutions to the relevant backward stochastic equations. In our case, and as it is in [5], the form of the required backward stochastic equation for this linear-type term is that of a mean-field backward stochastic differential equation—a form of equation that has been studied in [6] and [7].

In [21], the relevant quadratic-type term is also treated using an adjoint representation technique, and it is shown that the adjoint term can be given by the solution of another linear BSDE. We apply a comparable form of procedure in order to deal with the quadratic-type term in our problem. Although, the procedure is, to some extent, complicated by the mean-field nature of our problem. Here we have two forms of quadratic-type terms: those of a form

EP

hZ T r+

Y2,1(s)ζ^(r,)(s)² ds+Y2,1(T)ζ^(r,)(T)² i

for certain factorsY2,1; and those of a form ZT

r+

EP[Y2,2,1(s)ζ^(r,)(s)]EP[Y2,2,2(s)ζ^(r,)(s)] ds+EP[Y2,2,1(T)ζ^(r,)(T)]EP[Y2,2,2(T)ζ^(r,)(T)]

for certain factorsY2,2,1 andY2,2,2. A comparable situation exists in [5] also.

We treat the first form of quadratic-type term using an adjoint process that is defined as the solution of a certain BSDE, which, again, is similar to how the relevant first form of quadratic-type term is treated in [5], and related to how the relevant quadratic-type term is treated in [21]. We treat the second form of quadratic-type term by applying an adjoint representation procedure on the completion of a product probability space formed from (Ω,F,P) and a copy of itself. The relevant adjoint process is given by the solution of an equation that is of a class of backward stochastic equations that we introduce in this paper (see Section 4), and which we term conditional mean-field backward stochastic differential equations (CMFBSDEs). By using this third form of adjoint process we are able to show that the limit as↓0 of this second form of quadratic-type term multiplied by⁻¹, will be null. Thus these third-type adjoint processes do not appear in our final SMP equations. In [5] the authors establish bounds on their relevant second form of quadratic-type term, to imply that it is of ordero().

Define the shortened notation (using the notationu^∗_i(ui) as it is defined in equation (3)):

•

∂ψ(ω, t) :=∂ψ

ω, t, X^∗(ω, t), Z

Ω

φψ(ω⁰, t, X^∗(ω⁰, t), u^∗(ω⁰, t))P(dω⁰), u^∗(ω, t), ζ^(r,)(ω, t),

Z

Ω

∂φψ(ω⁰, t, X^∗(ω⁰, t), u^∗(ω⁰, t))ζ^(r,)(ω⁰, t)P(dω⁰)

∂⁽²⁾ψ(ω, t) :=∂⁽²⁾ψ

ω, t, X^∗(ω, t), Z

Ω

φψ(ω⁰, t, X^∗(ω⁰, t), u^∗(ω⁰, t))P(dω⁰), u^∗(ω, t), ζ^(r,)(ω, t),

Z

Ω

∂φψ(ω⁰, t, X^∗(ω⁰, t), u^∗(ω⁰, t))ζ^(r,)(ω⁰, t)P(dω⁰), Z

Ω

∂⁽²⁾φψ(ω⁰, t, X^∗(ω⁰, t), u^∗(ω⁰, t))ζ^(r,)(ω⁰, t)² P(dω⁰) for allψ∈ {b, σ, f1, . . . , fN}.

•

∂gi(ω) :=∂gi

ω, X^∗(ω, T), Z

Ω

φg_i(ω⁰, X^∗(ω⁰, T))P(dω⁰), ζ^(r,)(ω, T), Z

Ω

∂φg_i(ω⁰, X^∗(ω⁰, T))ζ^(r,)(ω⁰, T)P(dω⁰)

∂⁽²⁾gi(ω) :=∂⁽²⁾gi

ω, X^∗(ω, T), Z

Ω

φg_i(ω⁰, X^∗(ω⁰, T))P(dω⁰), ζ^(r,)(ω, T), Z

Ω

∂φg_i(ω⁰, X^∗(ω⁰, T))ζ^(r,)(ω⁰, T)P(dω⁰), Z

Ω

∂⁽²⁾φg_i(ω⁰, X^∗(ω⁰, T))ζ^(r,)(ω⁰, T)² P(dω⁰)

for alli∈ {1, . . . , N}.

•

∆^(r,)ψ(ω, t) :=ψ

ω, t, X^(r,)(ω, t), Z

Ω

φψ(ω⁰, t, X^(r,)(ω⁰, t), u^∗_i(ui)(ω⁰, t))P(dω⁰), u^∗_i(ui)(ω, t)

−ψ

ω, t, X^∗(ω, t), Z

Ω

φψ(ω⁰, t, X^∗(ω⁰, t), u^∗(ω⁰, t))P(dω⁰), u^∗(ω, t) δ^(r,)ψ(ω, t) :=ψ

ω, t, X^(r,)(ω, t), Z

Ω

φψ(ω⁰, t, X^(r,)(ω⁰, t), u^∗i(ui)(ω⁰, t))P(dω⁰), u^∗i(ui)(ω, t)

−ψ

ω, t, X^∗(ω, t), Z

Ω

φψ(ω⁰, t, X^∗(ω⁰, t), u^∗_i(ui)(ω⁰, t))P(dω⁰), u^∗_i(ui)(ω, t) δ^∗ψ(ω, t) :=ψ

ω, t, X^∗(ω, t), Z

Ω

φψ(ω⁰, t, X^∗(ω⁰, t), u^∗i(ui)(ω⁰, t))P(dω⁰), u^∗i(ui)(ω, t)

−ψ

ω, t, X^∗(ω, t), Z

Ω

φψ(ω⁰, t, X^∗(ω⁰, t), u^∗(ω⁰, t))P(dω⁰), u^∗(ω, t) for allψ∈ {b, σ, f1, . . . , fN}.

The families of all processesη^(r,)₁ , η^(r,)₂ , η_4,i^(r,), and random variablesη_3,i^(r,)are defined as in Lemma A.1.

Step 1. Decomposition.

By the definition of the MFBSDE (4), the result of Lemma A.1, and the Itˆo product formula then pi(T)ζ^(r,)(T) = pi(r+)ζ^(r,)(r+) +

Z T r+

pi(s)(∂b(s) +1

2∂⁽²⁾b(s) +η^(r,)₁ (s)) +qi(s)(∂σ(s) +1

2∂⁽²⁾σ(s) +η^(r,)₂ (s))

−ζ^(r,)(s) Z

Ω

hi(ω⁰, s, pi(s), pi(ω⁰, s), qi(s), qi(ω⁰, s))P(dω⁰) ds +

Z T r+

n pi(s)

∂σ(s) +1

2∂⁽²⁾σ(s) +η₂^(r,)(s)

+qi(s)ζ^(r,)(s)o dB(s) hence it follows, given the definition of the MFBSDE (4), that

EP

hZ T r+

∂fi(s)ds+∂gi

i

=EP

hZ T r+

∂fi(s)ds+pi(T)ζ^(r,)(T)i

=EP

h

pi(r+)ζ^(r,)(r+) + Z T

r+

ζ^(r,)(s)n

∂xfi(s) +∂xb(s)pi(s) +∂xσ(s)qi(s) +

Z

Ω

∂φfi(s)∂yfi(ω⁰, s) +∂φb(s)∂yb(ω⁰, s)pi(ω⁰, s) +∂φσ(s)∂yσ(ω⁰, s)qi(ω⁰, s)

−hi(ω⁰, s, pi(s), pi(ω⁰, s), qi(s), qi(ω⁰, s))

P(dω⁰)o ds +

Z T r+

pi(s)(1

2∂⁽²⁾b(s) +η₁^(r,)(s)) +qi(s)(1

2∂⁽²⁾σ(s) +η^(r,)₂ (s)) dsi

=EP

h

pi(r+)ζ^(r,)(r+) + Z T

r+

pi(s)(1

2∂⁽²⁾b(s) +η^(r,)₁ (s)) +qi(s)(1

2∂⁽²⁾σ(s) +η₂^(r,)(s)) dsi for allr∈[0, T) and∈(0, T −r].

Define, for eachr∈[0, T), the functionsI₁^(r), I₂^(r), I₃^(r), I₄^(r): (0, T −r]→Rby I₁^(r)() :=EP

hZ r+

r

∆^(r,)fi(s) dsi I₂^(r)() :=EP

h

pi(r+)ζ^(r,)(r+)i I₃^(r)() := 1

2EP

hZ T r+

pi(s)∂⁽²⁾b(s) +qi(s)∂⁽²⁾σ(s) +∂⁽²⁾fi(s)

ds+∂⁽²⁾gi

i

I₄^(r)() :=EP

hZ T r+

pi(s)η₁^(r,)(s) +qi(s)η₂^(r,)(s) +η_4,i^(r,)(s)

ds+η_3,i^(r,)i

for all∈(0, T−r], and so, from the above and Lemma A.1, then

Ji(u^∗i(u^(r,)_i ))−Ji(u^∗) =I₁^(r)() +I₂^(r)() +I₃^(r)() +I₄^(r)() for allr∈[0, T) and∈(0, T −r].

Step 2. Limit of⁻¹I₁^(r)().

By Assumption 2.1 and Lemma A.1, for anyr∈[0, T) then lim↓0⁻¹

Zr+

r

EP[δ^(r,)fi(s)] ds= 0.

Hence, by the fundamental theorem of calculus for the Lebesgue integral, lim

↓0⁻¹I₁^(r)() =EP[δ^∗fi(r)]

for Leb-almost allr∈[0, T].

Step 3. Limit of⁻¹I₂^(r)().

Using the Itˆo product formula, we have that I₂^(r)(r) =

Z r+

r

EP

h

pi(s)∆^(r,)b(s)

−Z

Ω

hi(ω⁰, s, pi(s), pi(ω⁰, s), qi(s), qi(ω⁰, s))P(dω⁰)

ζ^(r,)(s) +qi(s)∆^(r,)σ(s)i ds for allr∈[0, T) and ∈(0, T−r]. By the Cauchy-Schwarz inequality, Assumption 2.1, Lemma A.1, the properties of eachpi,qi, andhi, and the dominated convergence theorem, then

lim↓0⁻¹ Z r+

r

EP

h

pi(s)δ^(r,)b(s)

−Z

Ω

hi(ω⁰, s, pi(s), pi(ω⁰, s), qi(s), qi(ω⁰, s))P(dω⁰)

ζ^(r,)(s) +qi(s)δ^(r,)σ(s)i = 0 for anyr∈[0, T). Therefore, by the fundamental theorem of calculus for the Lebesgue integral,

lim↓0⁻¹I₂^(r)() = lim

↓0⁻¹ Zr+

r

EP[pi(s)δ^∗b(s) +qi(s)δ^∗σ(s)] ds=EP[pi(r)δ^∗b(r) +qi(r)δ^∗σ(r)]

for Leb-almost allr∈[0, T].

Step 4. Limit of⁻¹K₁^(r)(): Part 1.

Let

I₃^(r)() =K₁^(r)() +K₂^(r)() for

K₁^(r)() := 1 2EP

hZ T r+

n−Z^(r,)(s) +

∂x,x⁽²⁾fi(s) +EP[∂yfi(s)]∂⁽²⁾φf_i(s) +pi(s)∂x,x⁽²⁾b(s) +EP[pi(s)∂yb(s)]∂⁽²⁾φb(s)

+∂_x,x⁽²⁾σ(s)qi(s) +EP[∂yσ(s)qi(s)]∂⁽²⁾φσ(s)

Ξ^(r,)(s)o ds + (∂⁽²⁾_x,xgi+EP[∂ygi]∂⁽²⁾φg_i)Ξ^(r,)(T)i

and

K₂^(r)() := 1 2

ZT r+

n

EP[Z^(r,)(s)] +

2EP[ζ^(r,)(s)pi(s)∂x,y⁽²⁾b(s)]

+EP[ζ^(r,)(s)∂φb(s)]EP[pi(s)∂⁽²⁾y,yb(s)]

EP[∂φb(s)ζ^(r,)(s)]

+

2EP[ζ^(r,)(s)∂x,y⁽²⁾σ(s)qi(s)] +EP[ζ^(r,)(s)∂φσ(s)]EP[∂⁽²⁾y,yσ(s)qi(s)]

EP[∂φσ(s)ζ^(r,)(s)]

+

2EP[ζ^(r,)(s)∂_x,y⁽²⁾fi(s)] +EP[ζ^(r,)(s)∂φf_i(s)]EP[∂_y,y⁽²⁾fi(s)]

EP[∂φf_i(s)ζ^(r,)(s)]o ds +1

2

2EP[ζ^(r,)(T)∂_x,y⁽²⁾gi] +EP[ζ^(r,)(T)∂φg_i]EP[∂_y,y⁽²⁾gi]

EP[∂φg_iζ^(r,)(T)]

for allr∈[0, T) and∈(0, T−r], where Ξ^(r,):=ζ^(r,)(t)²and{Z^(r,): Ω×[0, T]→R|r∈[0, T), ∈ (0, T−r]}is, for eachr∈[0, T) and∈(0, T−r], defined by

Z^(r,)(t) :=Pi(t)

2ζ^(r,)(t)∂yb(t)EP[∂φb(t)ζ^(r,)(t)] + 2∂xσ(t)ζ^(r,)(t)∂yσ(t)EP[∂φσ(t)ζ^(r,)(t)]

+ (∂yσ(t)EP[∂φσ(t)ζ^(r,)(t)])²

+ 2Qi(t)ζ^(r,)(t)∂yσ(t)EP[∂φσ(t)ζ^(r,)(t)]

forP-almost allω∈Ω uniformly for allt∈[0, T].

Given the definition of the BSDE (5), one has, using the Itˆo product formula and Lemma A.1, that EP[(∂⁽²⁾x,xgi+EP[∂ygi]∂⁽²⁾φg_i)Ξ^(r,)(T)] =EP[Pi(T)Ξ^(r,)(T)]

=EP[Pi(r+)Ξ^(r,)(r+)] + Z T

r+

EP

h

R^(r,)₁ (s)−li(s, Pi(s), Qi(s))Ξ^(r,)(s) +Pi(s)

2ζ^(r,)(s)∂b(s) + (∂σ(s))²

+ 2Qi(s)ζ^(r,)(s)∂σ(s)i ds

where{R^(r,)₁ : Ω×[0, T]→R|r∈[0, T), ∈(0, T−r]}is defined, for eachr∈[0, T) and∈(0, T−r], as

R^(r,)₁ (t) := Pi(t)

2ζ^(r,)(t)1

2∂⁽²⁾b(t) +η₁^(r,)(t)

+n

∂σ(t) +1

2∂⁽²⁾σ(t) +η₂^(r,)(t)2

−(∂σ(t))²o + 2Qi(t)ζ^(r,)(t)1

2∂⁽²⁾σ(t) +η^(r,)₂ (t)

forP-almost allω∈Ω uniformly for allt∈[0, T]. Furthermore, it can be shown that

EP[Pi(T)Ξ^(r,)(T)] =EP

h

Pi(r+)Ξ^(r,)(r+) + Z T

r+

n

R^(r,)₁ (s) +Z^(r,)(s) +

−li(s, Pi(s), Qi(s)) + 2Pi(s)∂xb(s) +Pi(s)(∂xσ(s))²+ 2Qi(s)∂xσ(s)

Ξ^(r,)(s)o dsi and hence, given the definition of each`i, then

K₁^(r)() = 1 2EP

h

Pi(r+)Ξ^(r,)(r+) + Z T

r+

R^(r,)₁ (s) ds i

for allr∈[0, T) and∈(0, T −r]. Using Assumption 2.1, Lemma A.1 and the properties of eachPi

andQi, one may show that

lim↓0⁻¹EP

hZ T r+

R^(r,)₁ (s) dsi

= 0 for allr∈[0, T), and so

lim↓0⁻¹K₁^(r)() = lim

↓0

1 2⁻¹EP

h

Pi(r+)Ξ^(r,)(r+)i for allr∈[0, T).

Step 5. Limit of⁻¹K₁^(r)(): Part 2.

Using the Itˆo product formula, we have that EP[Pi(r+)Ξ^(r,)(r+)] =

Z r+

r

EP

h Pi(s)

2ζ^(r,)(s)∆^(r,)b(s) + (∆^(r,)σ(s))²

−`i(s, Pi(s), Qi(s))Ξ^(r,)(s) + 2Qi(s)ζ^(r,)(s)∆^(r,)σ(s) i

ds for allr∈[0, T) and ∈(0, T−r]. By the Cauchy-Schwarz inequality, Assumption 2.1, Lemma A.1, the properties of eachPi,Qi, and`i, Proposition 2.3, and the dominated convergence theorem, then

lim↓0⁻¹ Z r+

r

EP

h Pi(s)

2ζ^(r,)(s)∆^(r,)b(s) + (δ^(r,)σ(s))²+ 2δ^(r,)σ(s)δ^∗σ(s)

−`i(s, Pi(s), Qi(s))Ξ^(r,)(s) + 2Qi(s)ζ^(r,)(s)∆^(r,)σ(s)i ds= 0